Maxwell's equations predict that the speed of light is constant(proofs.wiki)
proofs.wiki
Maxwell's equations predict that the speed of light is constant
http://proofs.wiki/Maxwell%27s_equations_predict_that_the_speed_of_light_is_constant
11 comments
Exactly, it doesn't prove "constancy" of c unless you assume constancy of other constants. Even the isotropic and homogeneous assumption is incomplete because it doesn't make any assertions on values through time. Is it possible that u0 or E0 or c had a different value at very start of the big bang? It's a very tempting thought although professional physicists seem almost universally consider those ideas a giant crackpot.
I don't know why you are getting downvoted. I haven't had my morning coffee yet, so I am cynically going to presume a bunch of folks think you're nitpicking on the words, but you're not.
The derivation is correct, but it is not a proof of anything. Maxwell's equations aren't textbook PDEs that exist in a vacuum, devoid of any connection with the surrounding world. They are quantitative approximations of real-life phenomenae, that hold only under very specific circumstances, and break when those conditions (about e.g. distances) aren't met.
The proof that the speed of light in vacuum (which the article "helpfully" fails to mention) is constant is experimental. If the experiment would fail, Maxwell's equations would have to be modified. Any predictions they make are tested against the reality, not vice-versa.
Oh yes: I'm not nitpicking about their failure to mention it's the speed of light in vacuum. The final expression is constant only because \mu and \varepsilon are (supposed to be) constant in vacuum. Precisely the same expression predicts the (correct!) fact that the speed of light is not constant, but depends on the properties of the material it is travelling through.
Edit: this should be very obvious from the fact that the derivation is based entirely on the four laws that constitute Maxwell's equations, not on theorems. Laws are not deduced mathematically, but formulated by experiment. Their predictions cannot constitute a proof. It may be tempting to treat their predictions as such, because in mathematics any correct result obtained by derivation can constitute a proof, but this is not the way to look at a set of physical laws. Maxwell's equations also "prove" that things like squeezed light cannot exist -- despite the fact that it clearly does, since we can generate it. These incorrect predictions are not onthologically different from correct ones.
The derivation is correct, but it is not a proof of anything. Maxwell's equations aren't textbook PDEs that exist in a vacuum, devoid of any connection with the surrounding world. They are quantitative approximations of real-life phenomenae, that hold only under very specific circumstances, and break when those conditions (about e.g. distances) aren't met.
The proof that the speed of light in vacuum (which the article "helpfully" fails to mention) is constant is experimental. If the experiment would fail, Maxwell's equations would have to be modified. Any predictions they make are tested against the reality, not vice-versa.
Oh yes: I'm not nitpicking about their failure to mention it's the speed of light in vacuum. The final expression is constant only because \mu and \varepsilon are (supposed to be) constant in vacuum. Precisely the same expression predicts the (correct!) fact that the speed of light is not constant, but depends on the properties of the material it is travelling through.
Edit: this should be very obvious from the fact that the derivation is based entirely on the four laws that constitute Maxwell's equations, not on theorems. Laws are not deduced mathematically, but formulated by experiment. Their predictions cannot constitute a proof. It may be tempting to treat their predictions as such, because in mathematics any correct result obtained by derivation can constitute a proof, but this is not the way to look at a set of physical laws. Maxwell's equations also "prove" that things like squeezed light cannot exist -- despite the fact that it clearly does, since we can generate it. These incorrect predictions are not onthologically different from correct ones.
If I recall correctly, we measure ϵ0 and c experimentally, and then define µ0 accordingly.
That aside, is there any specific reason to believe homogeneity holds?
That aside, is there any specific reason to believe homogeneity holds?
Nowadays, in SI, they're defined as constants: c is exactly 299,792,458 m/s, µ_0 = 4π×10^−7 H/m, ε_0 = 1/(µ_0*c^2).
But these were chosen, based precise measurements, to ensure the (old) metre and the Ampere remained relatively unchanged.
As for your question, homogeneity is considered the simplest assumption you could make: you can't get simpler than a no-parameter model. It's also easy to disprove, if you can get data, making it a good scientific hypothesis.
There's some astronomers trying to measure the various fundamental constants in other regions of the galaxy via maser emissions. There was some interesting results a few years ago, but I don't think they're considered definitive. Still, lots of fame if they can do it!
But these were chosen, based precise measurements, to ensure the (old) metre and the Ampere remained relatively unchanged.
As for your question, homogeneity is considered the simplest assumption you could make: you can't get simpler than a no-parameter model. It's also easy to disprove, if you can get data, making it a good scientific hypothesis.
There's some astronomers trying to measure the various fundamental constants in other regions of the galaxy via maser emissions. There was some interesting results a few years ago, but I don't think they're considered definitive. Still, lots of fame if they can do it!
FYI, this is standard fare for any undergraduate-level E&M class.
The interesting part of deriving the wave equation from Maxwell's Laws, in my opinion, is how the permittivity and permeability of free space (epsilon_0 and mu_0) can both measured macroscopically in a small laboratory, by measuring forces between electric charges and current-carrying wires. Put these two seemingly unrelated values together and they give you the speed of light!
Another FYI - the author of the wiki proof in step 8 showed the vector calculus proof the 'long' (and boring) way, those vector calculus relations can usually be solved MUCH more easily and concisely using the Levi-Cevita algebra. http://en.m.wikipedia.org/wiki/Levi-Civita_symbol
The interesting part of deriving the wave equation from Maxwell's Laws, in my opinion, is how the permittivity and permeability of free space (epsilon_0 and mu_0) can both measured macroscopically in a small laboratory, by measuring forces between electric charges and current-carrying wires. Put these two seemingly unrelated values together and they give you the speed of light!
Another FYI - the author of the wiki proof in step 8 showed the vector calculus proof the 'long' (and boring) way, those vector calculus relations can usually be solved MUCH more easily and concisely using the Levi-Cevita algebra. http://en.m.wikipedia.org/wiki/Levi-Civita_symbol
Aside: I do computational applied math for a living, and haven't yet seen proofs.wiki before. Is it fairly new? An editable wiki just for formal proofs does seem like a decent idea.
the main page (http://proofs.wiki/index.php?title=Main_Page&action=history) is from December 5, 2014 (first revision is timestamped in 2011, but that is the MediaWiki default page)
http://proofs.wiki/index.php?title=Prerequisite:Copyrights&a... shows the copyright page is from December 6, 2014.
So, I assume this started around two months ago. The sparseness of content is not in conflict with that.
I think I would rather use a site that uses a formal proof checker behind it to verify committed theorems. It may get populated slower, but it can better be trusted.
http://proofs.wiki/index.php?title=Prerequisite:Copyrights&a... shows the copyright page is from December 6, 2014.
So, I assume this started around two months ago. The sparseness of content is not in conflict with that.
I think I would rather use a site that uses a formal proof checker behind it to verify committed theorems. It may get populated slower, but it can better be trusted.
It is a good idea!
Another excellent editable wiki for proofs has existed for some time:
https://www.proofwiki.org/wiki/Main_Page
It is nice to see growth in this fertile space.
I think it has to be fairly new (at least in its current location) just because the .wiki TLD is under a year old.
"[The top-level domain name]... became available to the general public on May 26, 2014"
- https://en.wikipedia.org/wiki/.wiki
"[The top-level domain name]... became available to the general public on May 26, 2014"
- https://en.wikipedia.org/wiki/.wiki
lawyeard, you seem to be dead.
It is a new wiki, it was created 3 months ago.
It's just too bad the speed of light is what it is. A few orders of magnitude higher and we've have a galaxy to explore. A few orders lower and we've have fun relativistic effects in daily life.
More likely that we just wouldn't be here...
Indeed. You'd think of all the people out there, the ones on HN would be a among the first to recognize an abstraction when they see one..
The "speed of light" isn't an intentional speed limit like the ones we have on highways. It's a mental cruth we use to describe a certain relationship that the elements forming our reality partake in .. Hence you can't reason about the "speed of light" having a different value the same way you could reason about a highway speed limit being altered in one direction or the other.
The "speed of light" isn't an intentional speed limit like the ones we have on highways. It's a mental cruth we use to describe a certain relationship that the elements forming our reality partake in .. Hence you can't reason about the "speed of light" having a different value the same way you could reason about a highway speed limit being altered in one direction or the other.
This is awesome. I could never prove anything, but I like to follow along (especially if the proof pertains to physics) and was thinking of presenting this in a cool way for people who might think science is lame or just never have been exposed (I got an undergraduate degree in Mechanical Engineering without knowing what Maxwell's eqns were). Looks like you have done a good job at it, I'll share with friends.
I always thought that the speed of light was sort-of built into Maxwell's equations through the permittivity and permeability of the medium (or vacuum), and comes out as the 1/sqrt of the product of these parameters. See [1].
Is this proof giving a deeper understanding, in the sense that these parameters somehow dissolve?
EDIT: Ok, I've actually read the article and it seems this proof is fairly standard, and was well known long before even the wikipedia page was written. It is nice that this proof is now available in such a clearly written form, though. I wish all the material on wikipedia was so clearly written :)
EDIT 2: It would be nice if this proof could be written down using differential forms, instead of the rather clumsy curl-div-grad notation, that gives rise to the big matrices in this proof. In fact, it bewilders me why the coordinate-dependent curl-div-grad notation is even used anymore in education nowadays, but that is another story.
[1] http://en.wikipedia.org/wiki/Maxwell%27s_equations
Is this proof giving a deeper understanding, in the sense that these parameters somehow dissolve?
EDIT: Ok, I've actually read the article and it seems this proof is fairly standard, and was well known long before even the wikipedia page was written. It is nice that this proof is now available in such a clearly written form, though. I wish all the material on wikipedia was so clearly written :)
EDIT 2: It would be nice if this proof could be written down using differential forms, instead of the rather clumsy curl-div-grad notation, that gives rise to the big matrices in this proof. In fact, it bewilders me why the coordinate-dependent curl-div-grad notation is even used anymore in education nowadays, but that is another story.
[1] http://en.wikipedia.org/wiki/Maxwell%27s_equations
Anyone who has taken a JD Jackson based course has derived this result. Special relativity is a fairly obvious consequence of E&M in hindsight.
FWIIW, this is a terrible proof with bad notation (you don't need to write out the matrix), which doesn't actually prove the interesting piece, which is the fact that the permittivity of free space is the same no matter what your frame of reference is.
FWIIW, this is a terrible proof with bad notation (you don't need to write out the matrix), which doesn't actually prove the interesting piece, which is the fact that the permittivity of free space is the same no matter what your frame of reference is.
I'm quite fond of differential forms, but the first time that they came up in my undergraduate math major was late in an optional second semester math analysis course (and even the first semester of of that course was more advanced than what a lot of undergraduate math programs seem to reach), and the treatment in that class was very brief.
I take it from your question that you think that there's a way to incorporate the differential forms formalism for electromagnetism into an undergraduate physics major at roughly a 2nd-year level (or maybe 3rd year). I'd be really curious to know what you have in mind: I teach physics professionally, and I have trouble imagining how to do forms justice in a way that matches the intuitive grounding of div/grad/curl (which many students already struggle with, mind you).
I take it from your question that you think that there's a way to incorporate the differential forms formalism for electromagnetism into an undergraduate physics major at roughly a 2nd-year level (or maybe 3rd year). I'd be really curious to know what you have in mind: I teach physics professionally, and I have trouble imagining how to do forms justice in a way that matches the intuitive grounding of div/grad/curl (which many students already struggle with, mind you).
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I mention in a separate thread but the 'clumsy' proof in step 8 is done much more cleanly using the Levi Cevita form.
Did you prove that the speed of light is constant in different inertial frames?
No:
"Note that there are two way to interpret this result:
The speed of light was constant with respect to the aether. This theory turned out to be false. Read more about the Luminiferous aether.
The theory of Relativity"
The most they say is just the assertion: "Maxwell's equations are Lorentz invariant"
Before the Michelson-Morley experiment 'disproved' it, it was assumed that the waves propagated through the 'aether', and that c was therefore the speed relative to the aether. So, in addition to Maxwell's equations, extra experimental data was needed.
"Note that there are two way to interpret this result:
The speed of light was constant with respect to the aether. This theory turned out to be false. Read more about the Luminiferous aether.
The theory of Relativity"
The most they say is just the assertion: "Maxwell's equations are Lorentz invariant"
Before the Michelson-Morley experiment 'disproved' it, it was assumed that the waves propagated through the 'aether', and that c was therefore the speed relative to the aether. So, in addition to Maxwell's equations, extra experimental data was needed.
Last sentence of the proof-- that's part of what Lorentz invariance is.
Wikipedia already has this at https://en.wikipedia.org/wiki/Electromagnetic_wave_equation
But it is not constant. It was much faster long ago. So there must be shome mishake.
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[1] homogeneity means that translating your experimental setup to a different location will not change the results of your experiment. Isotropy means that rotating your experimental setup will not change the results of your experiment.